rootof

RootOf is a symbolic representation used in computer algebra to denote a root of a polynomial without requiring an explicit closed form of the root. A RootOf object represents a specific root of a polynomial P(x) with rational or real coefficients, treated as an algebraic number in an abstract extension. It enables algebraic manipulation and reasoning about roots without solving P(x) explicitly, which is advantageous when roots are not expressible by radicals or when keeping the root symbolic simplifies computations.

Notation and indexing of RootOf varies by system. In many environments, RootOf(P, i) selects the i-th root of P according to a system-defined ordering, which may be based on numerical value, real vs. complex status, or another convention. The exact syntax and indexing are software-specific, but the core idea remains consistent: RootOf encodes a root as a constrained solution to P(x) = 0.

Properties and usage: RootOf objects satisfy the defining polynomial P(x) = 0, and they can be used in polynomial arithmetic, resultants, and solving steps within larger symbolic computations. They allow formulating inequalities, sign analysis, and root isolation without computing explicit root expressions. Numerical evaluation is typically available through a method that yields a floating-point approximation, such as an eval or n function, when a numeric value is needed.

Example and context: For a simple polynomial like P(x) = x^2 - 2, RootOf(P, i) denotes one of the two roots, with the specific root determined by the indexing convention. RootOf is widely used in educational contexts and in computational algebra systems to study the structure and properties of polynomial roots without resorting to explicit radical expressions.

See also: Algebraic number, Polynomial, Root isolation, Groebner basis, Resultant.